Reduced Order Modeling of Partial Differential Equations on Parameter-Dependent Domains Using Deep Neural Networks

TL;DR

This work addresses the challenge of solving PDEs with parameter-dependent domains by introducing DL-DA-ROM, a mesh-free reduced-order framework that learns a compact representation of the domain geometry from domain bitmaps via a convolutional autoencoder. By separating domain parametrization from the PDE solution surrogate, the approach enables accurate solution predictions across varying domain shapes, holes, and deformations, even when meshes are unavailable. The method demonstrates comparable accuracy to models using exact domain parameters in advection–diffusion and remains robust for nonlinear Navier–Stokes problems under moderate extrapolation and geometric perturbations. This domain-aware DL-ROM has practical implications for imaging-derived or measurement-based domains and offers a path toward efficient, three-dimensional extensions without requiring meshing or manual geometry alignment.

Abstract

Partial differential equations (PDEs) are widely used for modeling various physical phenomena. These equations often depend on certain parameters, necessitating either the identification of optimal parameters or the solution of the equations over multiple parameters. Performing an exhaustive search over the parameter space requires solving the PDE multiple times, which is generally impractical. To address this challenge, reduced order models (ROMs) are built using a set of precomputed solutions (snapshots) corresponding to different parameter values. Recently, Deep Learning ROMs (DL-ROMs) have been introduced as a new method to obtain ROM, offering improved flexibility and performance. In many cases, the domain on which the PDE is defined also varies. Capturing this variation is important for building accurate ROMs but is often difficult, especially when the domain has a complex structure or changes topology. In this paper, we propose a Deep-ROM framework that can automatically extract useful domain parametrization and incorporate it into the model. Unlike traditional domain parameterization methods, our approach does not require user-defined control points and can effectively handle domains with varying numbers of components. It can also learn from domain data even when no mesh is available. Using deep autoencoders, our approach reduces the dimensionality of both the PDE solution and the domain representation, making it possible to approximate solutions efficiently across a wide range of domain shapes and parameter values. We demonstrate that our approach produces parametrizations that yield solution accuracy comparable to models using exact parameters. Importantly, our model remains stable under moderate geometric variations in the domain, such as boundary deformations and noise - scenarios where traditional ROMs often require remeshing or manual adjustment.

Figures (14)

• Figure 1: Basic DL-ROM architecture: encoder $\Psi_E^S$ and decoder $\Psi_D^S$ are trained to minimize reconstruction error between full order model solution and its reconstruction, After training, encodings of solutions are calculated with encoder $\Psi_E^S$ and Multilayer perceptron (MLP) $\Phi_S$ is trained to map parameters to encoding. To obtain a solution for a new parameter $\boldsymbol{\lambda}$, parameters are passed through $\Phi_S$ and then through the decoder $\Psi_D^S$.
• Figure 2: Left panel shows a sample function, and the right panel shows its interpolation into an equidistant rectangular grid.
• Figure 3: Comparison between a mesh-based domain representation (left) and the corresponding interpolated characteristic function represented as an image used by our method (right). The binary image representation allows us to learn a low-dimensional domain encoding without requiring a mesh.
• Figure 4: Domain Autoencoder used for domain parametrization. The input to the convolutional neural network is a characteristic function represented as a bitmap image. The output of the autoencoder is a rectangular grid where each pixel value represents the predicted probability that the pixel belongs to the domain. The low-dimensional encoder output (latent representation) is used as the domain parametrization.
• Figure 5: Deep ROM architecture for PDEs with parameter-dependent domains. In addition to the standard Deep ROM structure, our method incorporates a domain autoencoder (bottom) that encodes the characteristic function of the domain into a low-dimensional latent representation. This representation is combined with other problem parameters and passed to a fully connected network $\Phi_S$, which predicts the solution encoding. The predicted latent solution is then decoded using $\Psi^S_D$ to reconstruct the solution on the full-order model (FOM) grid. Both the solution and domain autoencoders are trained using CNNs. The domain representation serves as a learned geometric parametrization that allows the ROM to generalize across different domain shapes and configurations.